Two families that cross at right angles

Fix two points. The ellipses with those foci are the curves on which the two distances add to a constant, one for each constant; the hyperbolas are the curves on which they differ by a constant. Both families fill the plane, and they cross.

Every crossing is a right angle. That is a surprising amount of order for two families defined by an addition and a subtraction, and the reason is the previous two rungs put side by side.

Why the angle is right #

Take a point $P$ and its two distances $r_1$ and $r_2$ to the foci. Exactly one ellipse of the family passes through it — the one whose constant is $r_1 + r_2$ — and exactly one hyperbola — the one whose constant is $|r_1 - r_2|$.

The rung below established that the ellipse’s tangent at $P$ makes equal angles with the two focal radii, on the reading that puts the radii on the same side; the rung above it established that the hyperbola’s tangent does the same on the reading that puts them on opposite sides. In one sentence: the ellipse’s tangent is the external bisector of the angle between the two focal radii and the hyperbola’s is the internal one.

The two bisectors of a pair of lines are perpendicular. That is elementary and it is the whole proof: the internal bisector splits the angle $\theta$ into two halves and the external one splits the supplement, so the angle between them is $\theta/2 + (\pi - \theta)/2 = \pi/2$, whatever $\theta$ is.

So the perpendicularity is not a property of conics at all. It is a property of bisectors, inherited by any two families whose tangents are the two bisectors of the same pair of directions.

The figure does not take that on trust. At every crossing it solves for the point exactly — the two equations give $x = aA/c$ and $y = bB/c$ in the two families’ semi-axes — and computes the gradients of the two implicit forms, and requires their inner product to vanish.

What a coordinate system is #

Two families of curves crossing at right angles, with exactly one of each through every point, is a coordinate system. Naming an ellipse and a hyperbola names a point — up to the reflections in the two axes, which is the same ambiguity polar coordinates have at the origin.

The familiar coordinate systems are exactly this. The horizontal and vertical lines are two families crossing at right angles, one of each through every point; the circles about the origin and the rays out of it are another, and the circles a map takes to circles are a third. Elliptic coordinates are a third, and they are the natural one for any question that treats two points as special.

The concrete form is worth having. Write $x = c\cosh\mu\cos\nu$ and $y = c\sinh\mu\sin\nu$. Fixing $\mu$ gives an ellipse and fixing $\nu$ gives a hyperbola, both with foci at $(\pm c, 0)$, and the perpendicularity is the statement that the two families of level curves of $\mu$ and $\nu$ are orthogonal.

That parameterisation makes the ambiguity explicit. $\nu$ and $-\nu$ give reflected points, and $\mu = 0$ collapses the ellipse to the segment between the foci — which is the coordinate system’s degenerate locus, in the way the origin is polar coordinates’.

It also shows what happens far from the foci, which is the reassuring limit. For large $\mu$ both $\cosh\mu$ and $\sinh\mu$ are close to $\tfrac12 e^{\mu}$, so $x$ and $y$ are close to a common multiple of $\cos\nu$ and $\sin\nu$ — the ellipses become circles and the hyperbolas become rays, and the system becomes the polar one. Elliptic coordinates are polar coordinates with the origin split into two points, and everything that is awkward about them is concentrated near the segment joining those two.

What an orthogonal family is for #

Orthogonality is not decoration. A coordinate system whose families cross at right angles is the only kind in which a great deal of machinery keeps its usual form, and the reason is the same one an orthonormal basis is worth building.

Distances stay separable. In an orthogonal system a small displacement’s length is a sum of squares of its components with no cross term, because the cross term is a product of two perpendicular directions and vanishes. In a skew system it does not, and every formula involving a length acquires an extra term.

Equations separate. The reason elliptic coordinates exist at all is that a differential equation on a region bounded by an ellipse can be solved by separating it into a function of $\mu$ times a function of $\nu$, and separation works precisely because the coordinates are orthogonal. The same is true of every classical coordinate system, and the list of coordinate systems in which the standard equations separate is finite and short — eleven of them in three dimensions, and this family is one.

And a conformal map produces them. The parameterisation above is $z = c\cosh w$, read as a complex function, and a complex-differentiable function takes perpendicular families to perpendicular families. That is the general source of orthogonal coordinate systems: take a conformal map and push the horizontal and vertical lines through it. Angles surviving a map while areas do not is the same property met in a different setting.

The last of those is the one that turns the theorem into a method. A confocal family is the image of a rectangular grid under one particular map; a different map gives a different orthogonal family; and every orthogonal coordinate system in the plane arises this way. So the question “which coordinate systems are there” becomes “which complex functions are there”, and the answer is a great many.

That also settles a question the picture raises and does not answer: why these two families rather than any other pair. The answer is that the pair is the image of the grid under the map $\cosh$, and $\cosh$ maps horizontal lines to ellipses and vertical lines to hyperbolas because it takes the imaginary axis round a circle and the real axis along a line, and combines the two multiplicatively. The family is a picture of one function, and the fact that both kinds of conic appear is a fact about that function rather than about conics.

The reflection reading of the same fact #

There is a second way to see the right angle, and it is worth having because it needs no bisectors and no derivatives.

Put a light source at one focus and a mirror along the ellipse through $P$. Every ray arrives at the other focus. Now put a mirror along the hyperbola through the same $P$ instead: a ray aimed at the far focus is turned towards the near one.

Consider the ray that leaves the first focus heading for $P$. Against the ellipse it reflects towards the second focus. Against the hyperbola it reflects — along the same incoming line, since it is aimed at neither focus, so that comparison does not run.

The comparison that does run is about the two mirrors’ normals. A mirror reflects a ray by flipping the component along its normal, and the ellipse’s normal at $P$ points into the region between the foci while the hyperbola’s points along it. Two mirrors through one point whose reflections of one ray differ by a half turn have perpendicular surfaces, and that is the statement again.

The value of the second reading is that it is physical. A surface that concentrates light at one point and a surface that relocates a focus can be placed at the same point of space, and the statement is that they are at right angles there — which is a fact an optical designer uses to reason about where a secondary mirror can sit.

The confocal family through one point #

A cleaner way to say what the coordinate system is: through every point of the plane (off the axes) there is exactly one ellipse and exactly one hyperbola of the family.

That is not obvious and it is provable in one line. Fix $P$ and consider the equation

$$\frac{x^2}{c^2 + \lambda} + \frac{y^2}{\lambda} = 1$$

as an equation for $\lambda$ with $x$ and $y$ the coordinates of $P$. Clearing denominators gives a quadratic in $\lambda$, so there are two roots; one is positive, giving an ellipse, and one lies between $-c^2$ and $0$, giving a hyperbola. One parameter, one quadratic, two roots, two curves.

That single family — one equation with one parameter producing both kinds — is the reason “confocal conics” is a family rather than two families. The ellipses are the members with $\lambda > 0$ and the hyperbolas those with $-c^2 < \lambda < 0$, and the segment between the foci is the boundary case $\lambda = 0$.

Reading it that way also explains the right angle a third time. Two members of a one-parameter family through the same point correspond to two roots of one quadratic, and the perpendicularity is a relation between those roots that comes out of the coefficients — which is the algebraic version of the bisector argument, and the version that generalises to three dimensions where there are three roots and three mutually perpendicular surfaces.

Where the ellipse touches the hyperbola #

The families interlock in a way that the picture shows and the algebra explains, and it is worth stating because it says how many crossings there are.

An ellipse with semi-axis $a$ and a hyperbola with semi-axis $A$ cross at four points, symmetric in both axes, provided $A < c < a$. If $A \ge c$ the hyperbola’s vertices lie outside the foci and the curve is not confocal with anything; if $a \le c$ the ellipse does not exist. The condition for a crossing is exactly the condition for both curves to be in the family, so every pair crosses and there is nothing to check.

The crossing point is $\left(aA/c,\; bB/c\right)$ where $b^2 = a^2 - c^2$ and $B^2 = c^2 - A^2$. Both formulas are symmetric in a way worth noticing: the ellipse contributes its longer axis and the hyperbola its shorter one to the $x$ coordinate, and the reverse to $y$.

Where it needs its conditions #

The two families must be confocal. Ellipses and hyperbolas with different foci cross at whatever angle they happen to; the orthogonality is a consequence of sharing the two points and of nothing else.

The degenerate members are not curves. The ellipse with $a = c$ is the segment joining the foci; the hyperbola with $A = c$ is the two rays outward along the axis; and the hyperbola with $A = 0$ is the perpendicular bisector. Those are the coordinate system’s boundary and the perpendicularity statement is vacuous on them.

The system is two-to-one. Two points symmetric about an axis have the same $(\mu, \nu)$ up to sign, so naming the two curves does not quite name the point. Every coordinate system built from level sets of two functions has an ambiguity somewhere, and this one’s is the reflections.

And “orthogonal” is a statement about tangents. Two curves crossing at a right angle means their tangents are perpendicular at the crossing, which is a local statement; nothing about the curves anywhere else is implied, and two curves can be perpendicular at one crossing and not at another in general. Here they are perpendicular at all four, by symmetry.

One further caution belongs with the coordinate reading. The system is orthogonal and it is not uniform: the curves crowd near the foci and spread out far away, so a small step in $\mu$ or $\nu$ corresponds to a physical distance that varies from place to place. Every calculation in these coordinates carries a scale factor for that reason, and forgetting it is the standard error. Orthogonality removes the cross terms and does not make the coefficients constant, and the two are frequently confused because in the Cartesian case both hold at once.

The three-gap theorem is a reminder from another field that a family of curves or points can be perfectly regular in one sense and thoroughly uneven in another, and the coordinate system here is regular in the sense that matters for separating an equation and uneven in the sense that matters for measuring one.

Where it came from #

Confocal conics are in Apollonius in the third century BC as a family of curves, and the orthogonality is much later — it is the kind of statement that needs a tangent to be a first-class object, which is a seventeenth-century development.

Their real career began with Jacobi in the 1830s, who used confocal quadrics — the three-dimensional version, where an ellipsoid, a hyperboloid of one sheet and one of two sheets pass through each point and meet pairwise at right angles — to integrate the equations of motion for a particle attracted to two fixed centres. The coordinate system was the tool that made the problem solvable — the same manoeuvre as rewriting a map in the basis its own directions provide — and it is a fair statement of what such systems are for: not a way of describing a region but a way of making an equation on it separate.

The same construction is why an ellipsoidal region is one of the few shapes on which the classical equations of physics can be solved exactly. There is a short list of such shapes and every one of them is the coordinate surface of one of the eleven separable systems.

What the pictures cannot show #

Four ellipses and three hyperbolas are drawn and both families are infinite. A picture of the coordinate system would need every curve, which is the plane; what is drawn is a grid, which is a sample of a system in the same way a piece of graph paper is a sample of the Cartesian one.

The right angle is checked at the twelve crossings the figure draws and is asserted at every point of the plane. The proof is the bisector argument, and it is prose — a picture of two bisectors being perpendicular would be a picture of the elementary fact rather than of the conics.

And the coordinate system’s use is not drawn at all. That an equation separates in these coordinates is a statement about differential operators, and no arrangement of curves on a page contains it.

The ladder from here #

Rungs above: the single sign that names the curve, where the general quadratic is classified and the family becomes an algebraic one. Confocal quadrics in three dimensions, where three surfaces meet pairwise at right angles at every point. Separation of variables in elliptic coordinates, and the Mathieu functions it produces. The Ivory and Graves theorems, which are further consequences of confocality and are as surprising as this one. And conics as projective objects, where all four kinds become one and the focal machinery has no meaning at all.

The property was already proved twice #

The habit is about what to do with two results that look like variations of each other.

The ellipse’s equal-angle property and the hyperbola’s are usually met as two theorems about two curves, proved separately and remembered separately. Written carefully they are one statement — the tangent bisects the angle between the focal radii — differing only in which of the two bisectors is meant.

The moment the two are written in the same words, the fact that the two bisectors are perpendicular is available for free, and this rung’s theorem falls out with no new work. It could not have fallen out of the two statements written in their usual separate forms, because those forms do not put the two tangents at the same point.

The general recommendation is to state a pair of parallel results in a way that makes the parallel explicit, even when the shared phrasing is slightly less natural for either one. The reward is that consequences of the pair become visible, and consequences of a pair are exactly what the separate statements hide.